Related papers: Turaev Surfaces

This is an expository note to give a brief review of classical elastica theory, mainly prepared for giving a more detailed proof of the author's Li--Yau type inequality for self-intersecting curves in Euclidean space. We also discuss some…

An expository description of smooth cubic curves in the real or complex projective plane.

Some elementary considerations are presented concerning Catenoids and their stability, separable minimal hypersurfaces, minimal surfaces obtainable by rotating shapes, determinantal varieties, minimal tori in S3, the minimality in Rnk of…

The Turaev genus of a knot is a topological measure of how far a given knot is from being alternating. Recent work by several authors has focused attention on this interesting invariant. We discuss how the Turaev genus is related to other…

We extend Turaev's theory of Euler structures and torsion invariants on 3-manifolds to the case of vector fields having generic behavior on the boundary. This allows to easily define gluings of Euler structures and to develop a completely…

This is an expository article on diagrammatic representations of knots and links in various settings via braids.

An extension of the Artin Braid Group with new operators that generate double and triple intersections is considered. The extended Alexander theorem, relating intersecting closed braids and intersecting knots is proved for double and triple…

A brief exposition of the point of higher topos theory in (mathematical) physics, commissioned for the Encyclopedia of Mathematical Physics 2nd ed.

This is an expository article for the Encyclopedia of Mathematical Physics on the subject in the title.

This is an expository paper which explores the ideas of the authors' paper "From Affine Geometry to Complex Geometry", arXiv:0709.2290. We explain the basic ideas of the latter paper by going through a large number of concrete, increasingly…

In earlier papers we introduced a representation of isotopy classes of compact surfaces embedded in the three-sphere by so called rectangular diagrams. The formalism proved useful for comparing Legendrian knots. The aim of this paper is to…

An exposition of the basic geometry of twistor integrals, intended for mathematicians.

This is the text of my Bourbaki seminar on the proof of the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.

This article will appear in the Encyclopedia of Mathematical Physics (Elsevier, 2006).

In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.

A mostly expository account of old questions about the relationship between polyhedra and topological manifolds. Topics are old topological results, new gauge theory results (with speculations about next directions), and history of the…

This paper has been withdrawn by the author; its content is properly cantained in the paper arXiv:0706.4447, entitled "Pure motives, mixed motives and extensions of motives associated to singular surfaces", and submitted on June 29, 2007.

Following Goussarov's paper `Interdependent Modifications of Links and Invariants of Finite Degree' [Topology 37 (1998) 595--602] we describe an alternative finite type theory of knots. While (as shown by Goussarov) the alternative theory…

The paper is a survey of known periodicity properties of finite type invariants of knots, and their applications.

This is an overview article on finite type invariants, written for the Encyclopedia of Mathematical Physics